The development of generative artificial intelligence (AI) and its associated tools has revolutionised the learning and use of foreign languages (L2). One such tool is machine translation (MT), which has become increasingly popular among university students worldwide, spurring research on MT use in L2 writing. However, previous research has primarily focused on the writing products of intermediate or advanced L2 learners, neglecting the writing process with MT of students with limited L2 proficiency. Therefore, this case study aimed to qualitatively explore how the Common European Framework of Reference for Languages (CEFR) A2 university students employ strategies for L2 writing with MT and how their strategies change after strategy instruction. Seven participants completed writing tasks on a PC before, immediately after, and four weeks after three one-hour out-of-class instruction sessions based on the Strategic Content Learning (SCL) approach. Their writing process was screen-recorded, followed by stimulated recall interviews to elicit their strategies, which were coded and categorised using a framework by O'Malley and Chamot (1990). The results showed an increase in students' elaborate use of strategies after instruction. In particular, strategy clusters were observed for all participants, demonstrating their cognitive engagement in the writing process. Furthermore, first-language (L1)-related strategies were used more frequently post-instruction, indicating learners' efforts to create translation-friendly L1 input for MT. The findings suggest that teaching MT-use strategies is crucial to fostering learners' active engagement in the L2 writing process in a technology-enhanced learning environment.